In this worksheet, we will practice finding the matrix of linear transformation which rotates every vector by a given angle and reflects them in the x- or y-axis.

**Q1: **

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in through an angle of and then reflects it across the -axis.

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**Q2: **

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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**Q3: **

A linear transformation is formed by rotating every vector in through an angle of , reflecting the resulting vector in the -axis, and finally reflecting this vector in the -axis. Find the matrix of this linear transformation.

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**Q4: **

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

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**Q5: **

A linear transformation is formed by reflecting every vector in across the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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**Q6: **

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

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**Q7: **

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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**Q8: **

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

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**Q9: **

Let the matrix represent rotation in the plane through an angle of and let the matrix represent reflection in the -axis.

What is the matrix ?

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What is the matrix ?

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What is the matrix ?

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**Q10: **

Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the -axis.

Find .

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Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the -axis and the line of reflection.

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What, therefore, is the slope of the line of reflection?

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If lies in the direction of the line of reflection, what is ?

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By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.

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**Q11: **

Consider the given figure.

The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

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Using the fact that is perpendicular to , find the equation of .

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Using the fact that , find the coordinates of and .

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Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

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**Q12: **

A linear transformation is formed by rotating every vector in through an angle of counterclockwise (when viewed from the positive -axis) about the -axis and then reflecting the resulting vector in the -plane. Find the matrix of this linear transformation.

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